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All right.
We're now on problem number 5, on page 408.
And they have this little drawing here, and it says, on
the disk shown above, a player spins the arrow, this little
blue arrow.
A player spins the arrow twice.
The fraction a/b is formed, where a is the number of the
sector where the arrow stopped after the first spin, and b is
a number of the sector where the arrow stopped after the
second spin.
On every spin, each of the numbered sectors has an equal
probability of being the sector on
which the arrow stops.
What is the probability that the fraction a/b is
greater than 1?
So they want to know the probability that a/b is
greater than 1.
Well, the first thing you need to realize is, this is just a
random number generator.
The probability that any number between 1 and 6 is
equal, so this could just be a roll of a six sided die.
And saying well, if I roll a six sided die twice, what's
the probability that the first roll is going to be larger
than the second roll, right?
Because in order for this fraction to be greater than 1,
the numerator has to be greater than the
denominator, right?
Or, you could just take a/b is greater than 1, and multiply
both sides by b.
And that's the same thing as the probability that a is
greater than b.
So we can set up a little table and we could say, well,
what's the probability of getting each of these numbers
on the first roll?
And then given that probability, or given that you
got, let's say, a 2 on the first roll, what are the odds
that the second roll is going to be less than the first
roll, right?
So let's say spin, call this a spin, 1, 2, 3, 4, 5, 6.
What's the probability of getting each of these spins?
I'm just making up notation as I go.
But, hopefully this is making sense to you.
So the probability of getting any one of
these is 1/6, right?
They all have an equal probability of happening.
1/6.
Now, given each of these, what's the probability that
soon. your second spin is going to be less than the
first spin?
This is the second spin, b.
Actually let's put an a here.
The probability of a.
That'll probably make more sense to you.
And then we want to know the probably that the second spin,
b, is less than a.
Well, if I got a 1 the first time, what's the probability
that I get a number less than 1 out of these?
Well, is there any number less than 1?
There's 1, but that's not even less than 1.
That's equal to 1, so there's a 0 chance that I get a number
less than 1, right?
Because there is no number less than 1 that I can get.
If I got a 2, well there's only one number less than 1
out of the six, right?
That's 1.
So there's a 1/6 chance.
A 3.
If I got a 3 on the first spin, I can only get a 1 or 2
on the second spin, right?
So that's a 2/6 chance.
2/6.
If I got a 4, I could get a 1, 2, or 3.
That's a 3/6.
I think you see the pattern here.
There's a 4/6 chance of getting something less than 5.
And there's a 5/6 chance of getting something less than 6.
And so, what's the combined probability now for each of
the scenarios?
So what is the probability that I get a 1, and then I get
something less than 1?
Well, you just have to multiply these probabilities.
So that's a 0.
What's the probability that I get a 2, and then something
less than 2?
It's 1/6 times 1/6 which is 1/36.
The probability that I get a 3, and then
something less than 3?
Multiply them.
It's 2/36.
I think you see the pattern here again.
3/36, 4/36, and then 5/36.
5/36 is the probability to get a 6, and then
something less than a 6.
So in general, the probability that I get my first spin is
greater than my second spin is going to be the sum of these
probabilities, right?
Because any one of these scenarios would satisfy a
being greater than b.
So I just add up these probabilities.
So the denominator for all of them is 36.
It says 0/36, right?
So it's essentially 1 plus 2 plus 3 plus 4 plus 5.
Now let me add carefully because this is where I
historically mess up my problems. 1 plus 2 is 3.
3 plus 3 is 6.
6 plus 4 is 10.
10 plus 5 is 15.
So that equals 15/36.
And that is choice you a.
Move on to the next problem.
We're on problem number 6.
Which of the following tables shows a relationship in which
w is proportional to x?
All right, so let's just write them all out.
So choice a, if we have wx, so they have 1, 3, 2, 4, 3, 5.
So w is proportional to x.
Well, that also means that x is going to be
proportional to w, right?
So in this case x is 3 times.
So here we have times 3.
0, 3.
Here we have times 2 and here we have times 5/3.
So we're multiplying by a different factor every time.
So w is not proportional to x and x is not
proportional to w.
I'm just trying to figure out what do I have to multiply w
by to get x?
What would I have to multiply x by to get w?
They all have to be the same in order for these two
variables to be proportional.
So choice a is not right.
Choice b.
w, x, 3/9.
Let's see what I'm doing, 3, 9, 4, 16, 5, 25.
It looks like, in every scenario here, we squared w to
get x or w is the square root of x.
But they're not proportional because here we
multiplied by 3.
Here we multiplied by 4, and here we multiplied by 5.
So once again, it's not going to be choice b.
c.
I will change colors for variety.
I changed it to brown.
C w, x, 5, 10, so I multiplied by 2.
This is what I'm multiplying by.
It's not 10 times 2.
6 to 18, I multiplied by 3.
I already know this is wrong, right?
I multiplied by two different factors.
It's not choice c.
Choice d.
w.
I did one of the later choices, just to make sure we
do a lot of work.
Choice d is 7 to 21.
I multiplied by 3 to do that.
8 to 24, times 3.
Looks good so far.
9 to 27, multiplied by 3
So, in every case, I multiplied by the same factor.
So w is proportionate to x and x is proportionate to w.
So d is our choice.
I will see you in the next video.
And we don't have to worry about e because we know that d
is our choice.
I'll see